Problem 49 - Thomas Calculus

Question

Show that increasing functions and decreasing functions are one-to-one. That is, show that for any \(x_{1}\) and \(x_{2}\) in \(I, x_{2} \neq x_{1}\) implies \(f\left(x_{2}\right) \neq f\left(x_{1}\right)\)

Step-by-Step Solution

Answer

Increasing and decreasing functions are one-to-one because they map distinct inputs to distinct outputs.

Step 1: Define an Increasing Function

A function \( f(x) \) defined on an interval \( I \) is called increasing if for every pair \( x_1, x_2 \) in \( I \), whenever \( x_1 < x_2 \), it follows that \( f(x_1) < f(x_2) \).

Step 2: Prove One-to-One for Increasing Functions

Suppose there exist two distinct points \( x_1 \) and \( x_2 \) in \( I \) such that \( x_1 < x_2 \). Since the function is increasing, \( f(x_1) < f(x_2) \). Therefore, for any distinct \( x_1 \) and \( x_2 \), \( f(x_2) \) cannot equal \( f(x_1) \), proving the function is one-to-one.

Step 3: Define a Decreasing Function

A function \( f(x) \) is called decreasing if for every pair \( x_1, x_2 \) in \( I \), whenever \( x_1 < x_2 \), it follows that \( f(x_1) > f(x_2) \).

Step 4: Prove One-to-One for Decreasing Functions

Suppose there exist two distinct points \( x_1 \) and \( x_2 \) in \( I \) such that \( x_1 < x_2 \). Since the function is decreasing, \( f(x_1) > f(x_2) \). Therefore, for any distinct \( x_1 \) and \( x_2 \), \( f(x_2) \) cannot equal \( f(x_1) \), proving the function is one-to-one.

Key Concepts

Increasing Functions

An increasing function is a type of mathematical function with a specific property. No matter which two points you pick on the function, if the first point is less than the second point, then the value of the function at the first point is also less than the value at the second point. This property ensures that as you move from left to right along the graph of the function, you are consistently moving upwards or staying level, but never going downwards.
The significance of increasing functions in mathematics, especially when it comes to one-to-one (injective) functions, is that they automatically satisfy the condition that no two different inputs have the same output.

• For example, imagine climbing a hill. If you start climbing at one point and keep climbing to another, higher point, you cannot be at the same height during your climb than at two different points.
• Mathematically, this means that if you have two distinct inputs, say \( x_1 \) and \( x_2 \), where \( x_1 < x_2 \), then the outputs will also be distinct, ensuring that \( f(x_1) < f(x_2) \).

This characteristic proves that an increasing function cannot map two different inputs to the same output, thus guaranteeing it is injective or one-to-one.

Decreasing Functions

Decreasing functions have a particular behavior opposite to that of increasing functions. For any two points where you have the first less than the second, the function's value at the first will be greater than its value at the second. In simpler terms, as you move from left to right on the graph of a decreasing function, you will consistently move downwards.
Just like increasing functions have their significance in mathematics, decreasing functions also prove to be one-to-one functions.

• Consider walking down a slope; you can't find yourself back at the same elevation between two different points without changing direction.
• If you take two different inputs, \( x_1 \) and \( x_2 \), with \( x_1 < x_2 \), then you'll find that \( f(x_1) > f(x_2) \).

This means that for a decreasing function, every different input produces a different output, making it injective. Basically, if the outputs are distinct, the function is one-to-one and doesn’t repeat any value for different domain elements.

Injective Functions

Injective functions, often simply called one-to-one functions, have a fascinating property. They are defined by the rule that if you take two different inputs, they must produce two different outputs. This means the function never assigns the same output to two different inputs.
The word "injective" comes from the notion of "injection," meaning each input uniquely determines its output.

• Imagine you have a collection of boxes (representing inputs) and keys (outputs). An injective function is like having a key that only fits into one particular box and no other.
• Mathematically speaking, if for all inputs \( x_1 \) and \( x_2 \), if \( f(x_1) = f(x_2) \) then it must mean \( x_1 = x_2 \). Otherwise, \( f(x_1) \) can't equal \( f(x_2) \) if \( x_1 eq x_2 \).

Both increasing and decreasing functions fall under this category. This is because they inherently ensure that no two different points on the graph can share the same vertical position, making the outputs unique for each input.